(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Analytical Mechanics"

308                      ANALYTICAL MECHANICS
path with a period equal to — .   The axes of the path coin-
co
cide with the coordinate axes.
Case IV.   When d ••= ± - and 6 = a; the path becomes a cir-
z>
cle, and the motion uniform circular motion with a period
1   4.      27T
equal to— -
CO
PROBLEMS.
Find the resultant motion due to the superposition of the motions defined by the following equations:
(1)   x — a cos o}t and y = a sin at.
(2)   x — a cos— (t + tQ)  and y = a sin ~~ t.
(3)   re = a sin coi and T/ = a cos — (£ + tQ).
(4)   a; = a sin (art + 5)  and y = a cos (art — 5).
(5)   £ = a cos art and ?/ = 6 sin art.
(6)   a; = a sin (art — 5) and y = 6 cos (art + 5).
(7)   x = a sin (art — ~) and y = 6 sin (ort + -)•
\        */                          \        ^/
(8)   £ = acos(coi + ™) and y = 6sin (coi — ™)-
\       «V                              \       «V
(9)   ^ = a cos (ut — j) and y = 6 cosf
(10)  .-r = a cos (art + 50 and ?/ = 6 sin (co^ + 5o).
241. Physical Pendulum. — Any rigid body which is free to oscillate under the action of its own weight is called a physical or a compound pendulum. Let A, Fig. 137, be a rigid body which is free to oscillate about a horizontal axis through the point 0 and perpendicular to the plane of the paper. Further let c denote the position of the center of mass and D its distance from the axis. Then the torque equation gives
/-— = — mgDsmd,                         (X)
at
where m is the mass of the body and 0 the angular displacement from the position of equilibrium.